Properties

Label 2-48e2-4.3-c2-0-44
Degree $2$
Conductor $2304$
Sign $i$
Analytic cond. $62.7794$
Root an. cond. $7.92334$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.46·5-s + 10i·7-s + 13.8i·11-s − 6.92·13-s − 30·17-s + 6.92i·19-s − 12i·23-s − 13.0·25-s − 51.9·29-s − 14i·31-s − 34.6i·35-s + 55.4·37-s − 6·41-s + 62.3i·43-s − 84i·47-s + ⋯
L(s)  = 1  − 0.692·5-s + 1.42i·7-s + 1.25i·11-s − 0.532·13-s − 1.76·17-s + 0.364i·19-s − 0.521i·23-s − 0.520·25-s − 1.79·29-s − 0.451i·31-s − 0.989i·35-s + 1.49·37-s − 0.146·41-s + 1.45i·43-s − 1.78i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $i$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1382835617\)
\(L(\frac12)\) \(\approx\) \(0.1382835617\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + 3.46T + 25T^{2} \)
7 \( 1 - 10iT - 49T^{2} \)
11 \( 1 - 13.8iT - 121T^{2} \)
13 \( 1 + 6.92T + 169T^{2} \)
17 \( 1 + 30T + 289T^{2} \)
19 \( 1 - 6.92iT - 361T^{2} \)
23 \( 1 + 12iT - 529T^{2} \)
29 \( 1 + 51.9T + 841T^{2} \)
31 \( 1 + 14iT - 961T^{2} \)
37 \( 1 - 55.4T + 1.36e3T^{2} \)
41 \( 1 + 6T + 1.68e3T^{2} \)
43 \( 1 - 62.3iT - 1.84e3T^{2} \)
47 \( 1 + 84iT - 2.20e3T^{2} \)
53 \( 1 - 17.3T + 2.80e3T^{2} \)
59 \( 1 - 62.3iT - 3.48e3T^{2} \)
61 \( 1 - 96.9T + 3.72e3T^{2} \)
67 \( 1 + 48.4iT - 4.48e3T^{2} \)
71 \( 1 - 60iT - 5.04e3T^{2} \)
73 \( 1 - 86T + 5.32e3T^{2} \)
79 \( 1 + 38iT - 6.24e3T^{2} \)
83 \( 1 - 13.8iT - 6.88e3T^{2} \)
89 \( 1 - 78T + 7.92e3T^{2} \)
97 \( 1 - 62T + 9.40e3T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.652116663742431685948823533364, −7.88691221591311851480868318730, −7.14668865259208752533276143248, −6.31992449616549241419187251319, −5.43130742508293413936809193277, −4.58006806654943098508296978420, −3.88251855168438108435987678511, −2.45365696188113506739933752585, −2.04072238375113022808650162438, −0.04365533444592473279095470707, 0.78868126268888265057129419050, 2.24964764112346689184497020241, 3.54512455862346784172102069312, 4.02133597114845622535346211034, 4.89226238519089421566075969861, 5.97562943403546011999093393079, 6.86950168518865883940903314179, 7.47562296393146571029787123377, 8.119958747255047620688117140295, 9.005166353829383169784800389866

Graph of the $Z$-function along the critical line